# Number Theory: Unique Numbers

## Main Question or Discussion Point

Does anyone know of a reference work that lists natural numbers with unique properties? Like 26, for example, being the only natural number sandwiched between a square (25) and a cube (27). Does such a reference book exist?

IH

Wikipedia has such information: http://en.wikipedia.org/wiki/38_(number [Broken])

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Thanx Micro, I was aware of certain Wikipedia articles; what I specifically am looking for though, is a systematic reference work of all know unique numbers. I could not find something resembling this on the net...

mfb
Mentor
It depends a lot on the things you consider as unique properties. Every number has unique properties, but most of them are boring ("is the only number x where x-23 and x-24 are primes" is another one for 26). Random collections are the best things you can find.

Hmm...is that a trivial uniqueness quality that you just mentioned for 26? Doesn't seem so to me but then I am the layman here...

Can one somehow 'define' mathematical triviality for such unique qualities I wonder...

IH

mfb
Mentor
It is trivial in the way that "x-23 prime and x-24 prime" requires two primes with a difference of just 1, and 2 and 3 are the only primes that satisfy this.
You can set this up for every integer.

Yes, if course...silly me...

Thanx Curious, exactly the type of thing I was looking for, thanx a million...I kept repeating "unique" in all my Google searches, so there you go...a little variety is always good...

IH

mfb
Mentor
Note that not all those entries are unique, and some of them just reflect our limited knowledge. And some are... pointless.

"151 is a palindromic prime." - true, but there are 7 smaller palindromic primes and probably infinitely more larger ones.
"146 = 222 in base 8." - so what?

Thanx for the clarification mob...funny I would have thought that a compendium of numbers with unique characteristics would be a given in number theory...quite surprised that it's so difficult to find...

IH